WMPVN Cellular Automata Rule Family

Outer totalistic cellular automata rules determine the next state of a cell from its current state and a count of the states of its immediately adjacent cells. On a two-dimensional rectangular cellular automata grid, the von Neumann neighbourhood (left) consists of the four orthogonally adjacent cells. The Moore neighbourhood (centre) consists of those plus the four diagonally adjacent cells. The WMPVN neighbourhood (right) counts the orthogonally adjacent cells twice and the diagonally adjacent cells once, producing a count range of 0-12 compared to 0-4 for von Neumann and 0-8 for Moore. Cellular automata rules are often expressed in terms of two lists of cell counts, one identifying the number of neighbours which allow a live cell to survive and the other the number that cause a dead/empty cell to be born. To keep the survive/born lists to strings of single characters, WMPVN defines a=10, b=11, c=12 in that context.

WMPVN also uses a concept of dying cells that was introduced implicitly for the Moore neigbourhood through Mirek Wójtowicz's Generations rule family whereby, when a previously live cell does not meet the survival criterion, it enters a dying state for a number of iterations where it neither looks at nor is counted by its neighbours. Technically, the dying state is implemented by running the cell through a series of numbered states, the third rule defining Geneerations/WMPVN parameter being the total number of such states, i.e. the number of iterations from live back to dead/empty plus two.

The core point of WMPVN was to provide a somewhat larger rule space to explore without sacrificing angular isotropy to the degree that can apply on a rectangular grid. Adding four to each of the survive/born ranges expands the nominal rule space by 28=256 times.

Early days

It turns out that rules which only vary by the presence or absence of one or more of the higher numbers in their survive and/or born strings are more likely to have similar mechanisms emerge, so there is a short hand form to describe a group of such rules which puts an x in the digit position where, counting back from the highest, a mechanism is the same across several rules. To date only Tony Smith and some other posters to ConwayLife.com's Other Cellular Automata forum have described any discoveries made in the WMPVN rule space. Smith is restricting himself to the 45678xx/45xxxx corner while ebcube reported some interesting results in 4568/456. Smith's corner is just one millionth of the total WMPVN rule space so there is plenty left for others.